Birkhoff–James ε–Orthogonality Sets for Matrices and Matrix polynomials

Stampfli and Williams in 1968 observed that the numerical range of a square nxn matrix A can be written as an infinite intersection of discs.

We observe the connection that this definition has with the notion of the Birkhoff–James ε–Orthogonality in a normed space and we use it to expand the idea of the numerical range to the case of rectangular matrices through defining the Birkhoff-James ε-orthogonality sets for rectangular matrices. We then study the basic properties of these sets.

Finally we take the new definition and apply it in the case of matrix polynomials in an effort to generalize the standard numerical range of matrix polynomials.

This is joint work with Panayiotis Psarrakos